Hyperelastic models are of particular interest in modeling biomaterials. In order to implement them, one must derive the stress and elasticity tensors from the given potential energy function explicitly. However, it is often cumbersome to do so because researchers in biomechanics may not be well-exposed to systematic approaches to derive the stress and elasticity tensors as it is vaguely addressed in literature. To resolve this, we present a framework of a general approach to derive the stress and elasticity tensors for hyperelastic models. Throughout the derivation we carefully elaborate the differences between formulas used in the displacement-based formulation and the displacement/pressure mixed formulation. Three hyperelastic models, Mooney–Rivlin, Yeoh and Holzapfel–Gasser–Ogden models that span from first-order to higher order and from isotropic to anisotropic materials, are served as examples. These detailed derivations are validated with numerical experiments that demonstrate excellent agreements with analytical and other computational solutions. Following this framework, one could implement with ease any hyperelastic model as user-defined functions in software packages or develop as an original source code from scratch.

中文翻译：

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导出超弹性各向同性和各向异性生物材料的应力和弹性张量的一般方法

超弹性模型在生物材料建模中特别受关注。为了实现它们，必须明确地从给定的势能函数中推导出应力和弹性张量。然而，这样做通常很麻烦，因为生物力学研究人员可能没有充分接触到系统方法来推导应力和弹性张量，因为它在文献中模糊地被提及。为了解决这个问题，我们提出了一个通用方法的框架，用于推导超弹性模型的应力和弹性张量。在整个推导过程中，我们仔细阐述了基于位移的公式和位移/压力混合公式中使用的公式之间的差异。三个超弹性模型，Mooney-Rivlin，从一阶到高阶以及从各向同性到各向异性材料的 Yeoh 和 Holzapfel-Gasser-Ogden 模型作为示例。这些详细的推导通过数值实验得到验证，这些实验证明了与分析和其他计算解决方案的出色一致性。遵循这个框架，人们可以轻松地将任何超弹性模型实现为软件包中的用户定义函数，或者从头开始开发为原始源代码。